We discuss an algorithm to compute transport maps that couple the uniform measure on $[0,1]^d$ with a specified target distribution $π$ on $[0,1]^d$. The primary objectives are either to sample from or to compute expectations w.r.t. $π$. The method is based on leveraging a polynomial surrogate of the target density, which is obtained by a least-squares or interpolation approximation. We discuss the design and construction of suitable sparse approximation spaces, and provide a complete error and cost analysis for target densities belonging to certain smoothness classes. Further, we explore the relation between our proposed algorithm and related approaches that aim to find suitable transports via optimization over a class of parametrized transports. Finally, we discuss the efficient implementation of our algorithm and report on numerical experiments which confirm our theory.